Algebra Examples

$f (x) = - 3^{2 x + 1} + 2$

Step 1

Set $- 3^{2 x + 1} + 2$ equal to $0$ .

$- 3^{2 x + 1} + 2 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $2$ from both sides of the equation.

$- 3^{2 x + 1} = - 2$

Step 2.2

Divide each term in $- 3^{2 x + 1} = - 2$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- 3^{2 x + 1} = - 2$ by $- 1$ .

$\frac{- 3^{2 x + 1}}{- 1} = \frac{- 2}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{3^{2 x + 1}}{1} = \frac{- 2}{- 1}$

Step 2.2.2.2

Divide $3^{2 x + 1}$ by $1$ .

$3^{2 x + 1} = \frac{- 2}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 2$ by $- 1$ .

$3^{2 x + 1} = 2$

Step 2.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (3^{2 x + 1}) = \ln (2)$

Step 2.4

Expand $\ln (3^{2 x + 1})$ by moving $2 x + 1$ outside the logarithm.

$(2 x + 1) \ln (3) = \ln (2)$

Step 2.5

Simplify the left side.

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Step 2.5.1

Simplify $(2 x + 1) \ln (3)$ .

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Step 2.5.1.1

Apply the distributive property.

$2 x \ln (3) + 1 \ln (3) = \ln (2)$

Step 2.5.1.2

Multiply $\ln (3)$ by $1$ .

$2 x \ln (3) + \ln (3) = \ln (2)$

Step 2.6

Move all the terms containing a logarithm to the left side of the equation.

$2 x \ln (3) + \ln (3) - \ln (2) = 0$

Step 2.7

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$2 x \ln (3) + \ln (\frac{3}{2}) = 0$

Step 2.8

Subtract $\ln (\frac{3}{2})$ from both sides of the equation.

$2 x \ln (3) = - \ln (\frac{3}{2})$

Step 2.9

Divide each term in $2 x \ln (3) = - \ln (\frac{3}{2})$ by $2 \ln (3)$ and simplify.

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Step 2.9.1

Divide each term in $2 x \ln (3) = - \ln (\frac{3}{2})$ by $2 \ln (3)$ .

$\frac{2 x \ln (3)}{2 \ln (3)} = \frac{- \ln (\frac{3}{2})}{2 \ln (3)}$

Step 2.9.2

Simplify the left side.

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Step 2.9.2.1

Cancel the common factor of $2$ .

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Step 2.9.2.1.1

Cancel the common factor.

$\frac{2 x \ln (3)}{2 \ln (3)} = \frac{- \ln (\frac{3}{2})}{2 \ln (3)}$

Step 2.9.2.1.2

Rewrite the expression.

$\frac{x \ln (3)}{\ln (3)} = \frac{- \ln (\frac{3}{2})}{2 \ln (3)}$

Step 2.9.2.2

Cancel the common factor of $\ln (3)$ .

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Step 2.9.2.2.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{- \ln (\frac{3}{2})}{2 \ln (3)}$

Step 2.9.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{3}{2})}{2 \ln (3)}$

Step 2.9.3

Simplify the right side.

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Step 2.9.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{3}{2})}{2 \ln (3)}$

Step 3